#1142508

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(2,3)(9,10,14,16)(11,13,12,15)(18,22)(20,23);(6,8)(9,11,14,12)(10,15,16,13);(1,2,4,3)(5,6)(7,8)(10,16)(11,12)(17,18)(19,23)(20,24)(21,22);(1,2,4,3)(9,12,14,11)(10,15,16,13)(17,19,21,24)(18,23,22,20);(1,3,4,2)(9,13,14,15)(10,11,16,12)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1049704);